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Question 1209810: John invested his money at a nominal interest rate of 12% compounded annually but the inflation rate at that time is equal to 5%. Determine the effective interest rate that John will receive considering the inflation rate.
Answer by CPhill(1959)   (Show Source):
You can put this solution on YOUR website! Let $P$ be the initial amount of money John invested.
The nominal interest rate is 12% compounded annually. So, after one year, the amount of money John has is:
$A = P(1 + 0.12) = 1.12P$.
The inflation rate is 5%. This means that the purchasing power of money decreases by 5% per year.
Let's consider the purchasing power of the money John has after one year.
If there were no inflation, John would have $1.12P$.
However, due to inflation, the value of money has decreased.
To determine the real value of the money, we need to adjust for the inflation.
Let $r$ be the effective interest rate that John receives considering the inflation rate.
Then $P(1+r)$ is the real value of the investment after one year in terms of the initial purchasing power.
We want to find $r$ such that $P(1+r)$ is equivalent to $1.12P$ after accounting for a 5% inflation rate.
$P(1+r)(1.05) = 1.12P$ is incorrect.
The initial purchasing power is $P$. After one year, John has $1.12P$. The inflation is 5%, so the price of goods increases by 5%.
Let $V_0$ be the initial value of goods that John can buy with $P$. After one year, the price of the same goods is $1.05V_0$.
John has $1.12P$ after one year.
The purchasing power after one year is $\frac{1.12P}{1.05P} = \frac{1.12}{1.05} = \frac{112}{105} = \frac{16}{15}$.
Let $r$ be the effective interest rate. Then $1+r = \frac{1.12}{1.05}$.
$r = \frac{1.12}{1.05} - 1 = \frac{112-105}{105} = \frac{7}{105} = \frac{1}{15}$.
$r = \frac{1}{15} \approx 0.066666...$
$r \approx 6.6667\%$.
Alternatively, we can use the formula:
Effective interest rate $\approx$ Nominal interest rate - Inflation rate.
$12\% - 5\% = 7\%$.
However, this is an approximation.
The exact formula is:
$1 + r = \frac{1 + \text{Nominal interest rate}}{1 + \text{Inflation rate}}$.
$1 + r = \frac{1.12}{1.05}$
$r = \frac{1.12}{1.05} - 1 = \frac{1.12 - 1.05}{1.05} = \frac{0.07}{1.05} = \frac{7}{105} = \frac{1}{15} = 0.06666...$
$r = 6.666...\%$.
Final Answer: The final answer is $\boxed{6.6667}$
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